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MinimalPrimes :: minprimes

minprimes -- minimal primes in a polynomial ring over a field

Synopsis

Description

Given an ideal in a polynomial ring, or a quotient of a polynomial ring whose base ring is either QQ or ZZ/p, return a list of minimal primes of the ideal.

i1 : R = ZZ/32003[a..e]

o1 = R

o1 : PolynomialRing
i2 : I = ideal"a2b-c3,abd-c2e,ade-ce2"

             2     3           2              2
o2 = ideal (a b - c , a*b*d - c e, a*d*e - c*e )

o2 : Ideal of R
i3 : C = minprimes I;
i4 : netList C

     +---------------------------+
o4 = |ideal (c, a)               |
     +---------------------------+
     |              2     3      |
     |ideal (e, d, a b - c )     |
     +---------------------------+
     |ideal (e, c, b)            |
     +---------------------------+
     |ideal (d, c, b)            |
     +---------------------------+
     |ideal (d - e, b - c, a - c)|
     +---------------------------+
     |ideal (d + e, b - c, a + c)|
     +---------------------------+
i5 : C2 = minprimes(I, Strategy=>"NoBirational", Verbosity=>2)
  Strategy: Linear            (time .00150707)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000614)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00373436)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00415143)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0464025)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .007601)   #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00448535)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00341463)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000602982)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0002967)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000300164)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00190717)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00468624)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00498521)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00354058)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00221811)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00615438)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00317868)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00280771)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00337534)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000016281)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000042888)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000028269)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000025331)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000055948)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000013194)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00152856)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000057718)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000034609)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000330027)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000349437)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000981)   #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0011239)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000173903)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000179188)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000304111)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000268872)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00120674)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00188217)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000013729)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000016658)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .00002911)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000019909)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00777406
#minprimes=6 #computed=10

                                  2     3
o5 = {ideal (c, a), ideal (e, d, a b - c ), ideal (e, c, b), ideal (d, c, b),
     ------------------------------------------------------------------------
     ideal (d - e, b - c, a - c), ideal (d + e, b - c, a + c)}

o5 : List
i6 : C1 = minprimes(I, Strategy=>"Birational", Verbosity=>2)
  Strategy: Linear            (time .00153967)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000085371)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00331872)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00462256)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00900692)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00312665)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00253913)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00273131)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000661531)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000490969)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000344523)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00251723)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00231943)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00370517)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0034861)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00313035)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00360205)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00230812)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00240624)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00284902)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000014342)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00004528)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011155)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000038123)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000061019)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001559)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00144489)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000055653)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000038086)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000573214)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000262947)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0015875)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00108851)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000173738)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000140845)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000301908)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000591642)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00142595)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00294717)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000018335)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000038252)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00582085)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00812966)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000393071)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000235532)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000058633)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00005297)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001529)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000016024)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0102427
#minprimes=6 #computed=10

                                  2     3
o6 = {ideal (c, a), ideal (e, d, a b - c ), ideal (e, c, b), ideal (d, c, b),
     ------------------------------------------------------------------------
     ideal (d - e, b - c, a - c), ideal (d + e, b - c, a + c)}

o6 : List

Caveat

This will eventually be made to work over GF(q), and over other fields too.

Ways to use minprimes :